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Solving Schrödinger Bridges via Maximum Likelihood
The Schrödinger bridge problem was proposed in the 1930s by Erwin Scrödinger. It involves two probability distributions at a start time an end time. They are related by a dynamic process. Deciding the form of that dynamic process given these distributions is a challenging problem. In this work we provide a maximum likelihood approach to solving it.
The Schrödinger bridge problem (SBP) finds the most likely stochastic evolution between two probability distributions given a prior stochastic evolution. As well as applications in the natural sciences, problems of this kind have important applications in machine learning such as dataset alignment and hypothesis testing. Whilst the theory behind this problem is relatively mature, scalable numerical recipes to estimate the Schrödinger bridge remain an active area of research. Our main contribution is the proof of equivalence between solving the SBP and an autoregressive maximum likelihood estimation objective. This formulation circumvents many of the challenges of density estimation and enables direct application of successful machine learning techniques. We propose a numerical procedure to estimate SBPs using Gaussian process and demonstrate the practical usage of our approach in numerical simulations and experiments.